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A connected graph G is said to be z-homogeneous if any isomorphism between finite connected induced subgraphs of G extends to an automorphism of G. Finite zhomogeneous graphs were classified in [17]. We show that z-homogeneity is equivalent to finite-transitivity on the class of infinite locally finite graphs. Moreover, we classify the graphs satisfying these properties. Our study of bipartite z-homogeneous graphs leads to a new characterization for hypercubes. Homogeneous GraphsWe study various notions of homogeneity for graphs. We begin by fixing some terminology that will be used throughout this article. The reader may consult [12] for undefined graph theoretic terms that appear here.We view each graph G as a set of vertices with a binary edge relation that is both symmetric and irreflexive. In particular, all graphs in this paper are undirected graphs with neither loops nor multiple edges. If the edge relation holds for vertices a and b of graph G, then we refer to the set {a, b} as an edge of G and say that a and b are adjacent in G or a and b are neighbors in G. We denote the set of vertices and the set of edges by V (G) and E(G), respectively.A path of length r from a to b in a graph G is a sequence of r+1 distinct vertices starting from a and ending with b such that consecutive vertices are adjacent in G. We speak of an (a, b)-path if we want to emphasis the start Mathematics Subject Classification (2000): 05C75

We identify the locally finite graphs that are quantifier-eliminable and their first order theories in the signature of distance predicates.Our study of quantifier-eliminable locally finite graphs was motivated by the results in [4,11]. The authors of [11] showed that the theory of the complete binary tree admits quantifier elimination in the signature L ∞ consisting of distance predicates. Their proofs are essentially syntactic. This prompted us to look for a proof closer in spirit to Robinson's message: to prove quantifier-elimination, use model theory rather than syntactic methods whenever possible. 1 We ended up answering the question of which locally finite graphs are quantifiereliminable in the signature L ∞ . They are precisely the 6-transitive graphs determined by Cameron in [2] and the infinite locally finite graphs with each connected component isomorphic to a member of a class of graphs that we call clique-trees. As a consequence of this result, we also show that complete rooted trees are quantifiereliminable in L ∞ ∪ {r} where r is the constant symbol for the root. This generalizes the corresponding result in [11] about the complete binary tree. Finitely transitive locally finite graphsWe summarize here the results from graph theory that we need. The reader can consult [6] for the graph theoretic terms that appear subsequently. Graphs in this article have neither loops nor multiple edges. A graph is locally finite if each vertex has finite valency. In particular, finite graphs are locally finite. We now describe a family of infinite locally finite graphs. Given natural numbers m, n > 1, consider a graph with the following four properties.-The graph is connected.-Each vertex has degree at most (m − 1)n.-Each vertex is the intersection of n distinct m-cliques. That is, given any vertex v there exist n disjoint sets of vertices D 1 , D 2 , . . . , D n so that for each i, |D i | = m − 1 and the induced subgraph on D i ∪ {v} is an m-clique. Moreover, there are no edges between D i and D j for i = j.-For n ≥ 4, no induced subgraph is an n-cycle.Note that, with the exception of connectedness, these properties are all elementary. It is easy to construct an isomorphism between any two graphs possessing all four of these properties. We call the unique (up to isomorphism) graph with these properties the (m, n) clique-tree. A clique-tree is simply an (m, n) clique-tree for some m, n.

We consider various extensions of first-order logic. Informally, a logic 𝓛 is an extension of first-order logic if every sentence of first-order logic is also a sentence of 𝓛. We also require that 𝓛 is closed under conjunction and negation and has other basic properties of a logic. In Section 9.4, we list the properties that formally define the notion of an extension of first-order logic. Prior to Section 9.4, we provide various natural examples of such extensions. In Sections 9.1–9.3, we consider, respectively, second-order logic, infinitary logics, and logics with fixed-point operators. We do not provide a thorough treatment of any one of these logics. Indeed, we could easily devote an entire chapter to each. Rather, we define each logic and provide examples that demonstrate the expressive power of the logics. In particular, we show that none of these logics has compactness. In the final Section 9.4, we prove that if a proper extension of first-order logic has compactness, then the Downward Löwenhiem–Skolem theorem must fail for that logic. This is Lindstrom’s theorem. The Compactness theorem and Downward Löwenheim–Skolem theorem are two crucial results for model theory. Every property of first-order logic from Chapter 4 is a consequence of these two theorems. Lindström’s theorem implies that the only extension of first-order logic possessing these properties is first-order logic itself. Second-order logic is the extension of first-order logic that allows quantification of relations. The symbols of second-order logic are the same symbols used in first-order logic. The syntax of second-order logic is defined by adding one rule to the syntax of first-order logic. The additional rule makes second-order logic far more expressive than first-order logic. Specifically, the syntax of second-order logic is defined as follows. Any atomic first-order formula is a formula of second-order logic. Moreover, we have the following four rules: (R1) If φ is a formula then so is ￢φ. (R2) If φ and ψ are formulas then so is φ ∧ ψ. (R3) If φ is a formula, then so is ∃x φ for any variable x.

We continue our study of Model Theory. This is the branch of logic concerned with the interplay between sentences of a formal language and mathematical structures. Primarily, Model Theory studies the relationship between a set of first-order sentences T and the class Mod(T) of structures that model T. Basic results of Model Theory were proved in the previous chapter. For example, it was shown that, in first-order logic, every model has a theory and every theory has a model. Put another way, T is consistent if and only if Mod(T) is nonempty. As a consequence of this, we proved the Completeness theorem. This theorem states that T ├ φ if and onlyif M ╞ φ for each M in Mod(T). So to study a theory T, we can avoid the concept of ├ and the methods of deduction introduced in Chapter 3, and instead work with the concept of ╞ and analyze the class Mod(T). More generally, we can go back and forth between the notions on the left side of the following table and their counterparts on the right. Progress in mathematics is often the result of having two or more points of view that are shown to be equivalent. A prime example is the relationship between the algebra of equations and the geometry of the graphs defined by the equations. Combining these two points of view yield concepts and results that would not be possible in either geometry or algebra alone. The Completeness theorem equates the two points of view exemplified in the above table. Model Theory exploits the relationship between these two points of view to investigate mathematical structures. First-order theories serve as our objects of study in this chapter. A first-order theory may be viewed as a consistent set of sentences T or as an elementary class of structures Mod(T). We shall present examples of theories and consider properties that the theories mayor may not possess such as completeness, categoricity, quantifier-elimination, and model-completeness. The properties that a theory possesses shed light on the structures that model the theory. We analyze examples of first-order structures including linear orders, vector spaces, the random graph, and the complex numbers.